What Are The Zeros Of A Quadratic Polynomial?
The zeros of a quadratic polynomial $ax^2 + bx + c$ (with $a \neq 0$) are the values of $x$ that make the whole expression equal to zero. A zero is simply an input that sends the output to $0$. Because the highest power is $2$, a quadratic has at most two zeros, and those zeros are also called its roots or solutions.
A quadratic polynomial is a polynomial whose highest power is two, written in the standard form $ax^2 + bx + c$. The letters $a$, $b$, and $c$ are fixed numbers called coefficients, and $a$ cannot be $0$, otherwise the $x^2$ term vanishes and the expression is no longer quadratic. Setting it equal to zero turns it into a quadratic equation, and the solutions of that equation are exactly the zeros of the polynomial.
Take the ball from a moment ago. If its height is a quadratic in time, the ground is where the height reads $0$, so the launch and the landing are the two zeros. Everything below is a way to find those two values, or to describe them.
How Do You Find The Zeros Of A Quadratic Polynomial?
There are three standard ways to find the zeros of a quadratic polynomial: factoring, the quadratic formula, and completing the square. Factoring is quickest when it works, the formula always works, and completing the square shows why the formula is true. Pick whichever fits the numbers in front of you.
How Do You Find Zeros By Factoring?
Factoring rewrites the quadratic as a product of two linear factors, then uses the fact that a product is zero only when one of its factors is zero.
Example 1: Find the zeros of $x^2 - 5x + 6$.
Set the polynomial equal to zero, then split the middle term into two numbers that multiply to $6$ and add to $-5$:
$$x^2 - 5x + 6 = 0$$
$$(x - 2)(x - 3) = 0$$
By the zero product property, each factor can be set to zero on its own:
$$x - 2 = 0 \quad \Rightarrow \quad x = 2$$
$$x - 3 = 0 \quad \Rightarrow \quad x = 3$$
Final answer: the zeros are $x = 2$ and $x = 3$. For more cases, see factorization of quadratic polynomials.
How Do You Use The Quadratic Formula?
When a quadratic does not factor neatly, the quadratic formula finds the zeros directly from the coefficients:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Example 2: Find the zeros of $2x^2 + x - 6$.
Here $a = 2$, $b = 1$, and $c = -6$. Substitute those values, being careful with the signs:
$$x = \frac{-1 \pm \sqrt{1^2 - 4(2)(-6)}}{2(2)} = \frac{-1 \pm \sqrt{49}}{4} = \frac{-1 \pm 7}{4}$$
The $\pm$ splits into two values:
$$x = \frac{-1 + 7}{4} = \frac{3}{2} \qquad x = \frac{-1 - 7}{4} = -2$$
Final answer: the zeros are $x = \tfrac{3}{2}$ and $x = -2$. The same values, viewed from the equation side, are covered in roots of a quadratic equation.
How Does Completing The Square Work?
Completing the square rebuilds the quadratic into a perfect square plus a constant, which isolates $x$ even when no clean factors exist.
Example 3: Find the zeros of $x^2 - 6x + 7$, which does not factor over the integers.
$$x^2 - 6x + 7 = 0$$
$$x^2 - 6x = -7$$
Add $\left(\tfrac{6}{2}\right)^2 = 9$ to both sides so the left side becomes a perfect square:
$$x^2 - 6x + 9 = 2$$
$$(x - 3)^2 = 2$$
$$x - 3 = \pm\sqrt{2}$$
Final answer: the zeros are $x = 3 + \sqrt{2}$ and $x = 3 - \sqrt{2}$. The full method sits at completing the square.
What Does The Discriminant Tell You About The Zeros?
The discriminant is the expression $b^2 - 4ac$, the part under the square root in the quadratic formula, and its sign alone tells you how many real zeros a quadratic polynomial has. You can read off the number of zeros before doing any solving. A fuller treatment lives at discriminant.
Table: What the discriminant reveals about the number of real zeros.
Discriminant $b^2 - 4ac$ | Number of real zeros | What the graph does |
|---|---|---|
Positive ($> 0$) | Two distinct real zeros | Parabola crosses the x-axis twice |
Zero ($= 0$) | One repeated real zero | Parabola touches the x-axis once, at the vertex |
Negative ($< 0$) | No real zeros (two complex) | Parabola never meets the x-axis |
A negative discriminant does not mean you made a mistake. It means the parabola sits entirely above or entirely below the x-axis, so the two zeros are complex numbers rather than real ones.
How Are The Zeros Related To The Coefficients?
You can learn a great deal about the zeros without ever finding them. For a quadratic polynomial $ax^2 + bx + c$ with zeros $\alpha$ and $\beta$, the sum of the zeros is $-\tfrac{b}{a}$ and the product of the zeros is $\tfrac{c}{a}$.
To see where those come from, write the polynomial in its factored form. If $\alpha$ and $\beta$ are the zeros, then:
$$a(x - \alpha)(x - \beta) = ax^2 - a(\alpha + \beta)x + a\alpha\beta$$
Matching this with $ax^2 + bx + c$ term by term gives $b = -a(\alpha + \beta)$ and $c = a\alpha\beta$, which rearrange into the two relationships:
$$\alpha + \beta = -\frac{b}{a} \qquad \alpha\beta = \frac{c}{a}$$
Table: The relationship between the zeros and the coefficients of $ax^2 + bx + c$.
Quantity | Formula | For $2x^2 + x - 6$ |
|---|---|---|
Sum of zeros ($\alpha + \beta$) | $-\dfrac{b}{a}$ | $-\dfrac{1}{2}$ |
Product of zeros ($\alpha\beta$) | $\dfrac{c}{a}$ | $-3$ |
Run the link backwards and you can build a quadratic from its zeros. A polynomial with zeros $\alpha$ and $\beta$ can be written as $x^2 - (\alpha + \beta)x + \alpha\beta$, so zeros of $2$ and $3$ rebuild $x^2 - 5x + 6$.
What Do The Zeros Look Like On A Graph?
The real zeros of a quadratic polynomial are the x-coordinates where its graph, a parabola, crosses the x-axis. Two real zeros give two crossings, a repeated zero gives a single touch at the vertex of the parabola, and no real zeros means the curve never reaches the axis. The axis of symmetry sits exactly halfway between the two zeros, at $x = -\tfrac{b}{2a}$.
For more on the shape itself, see parabola.
Why Do The Zeros Of A Quadratic Polynomial Matter?
A zero answers a question that comes up constantly: when does this quantity reach nothing? That question turns out to hide inside three very different jobs.
They solve the equation. Finding the zeros of $ax^2 + bx + c$ is the same task as solving $ax^2 + bx + c = 0$, one of the most common problems in all of algebra.
They bridge algebra and geometry. Each real zero is a point where the parabola meets the x-axis, so an algebra problem you solve on paper becomes a picture you can see.
They rebuild the whole polynomial. Once you know both zeros, the factored form $a(x - \alpha)(x - \beta)$ recovers the entire expression, so two numbers carry all of its information.
So much of a quadratic's behaviour, its graph, its factors, and its solutions, traces back to those two values. That is also why the idea extends: the same question drives the study of zeros of a polynomial of any degree.
Who Discovered The Link Between Zeros And Coefficients?
The rule that the sum of the zeros is $-\tfrac{b}{a}$ and their product is $\tfrac{c}{a}$ has a name, and a person behind it: the French mathematician François Viète.
That link is now called Viète's formulas, and it is exactly the sum and product relationship used earlier in this article. Two other mathematicians shaped how we solve quadratics in the first place:
Brahmagupta (598–670 CE, India) gave one of the first general methods for solving quadratic equations, and treated zero and negative numbers as ordinary quantities when much of the world still avoided them. His MacTutor biography covers both.
Al-Khwarizmi (around 780–850 CE, Baghdad) set out a systematic way to solve quadratics by completing the square, and the word "algebra" comes from al-jabr in the title of his book.
Where Are The Zeros Of A Quadratic Polynomial Used In The Real World?
The two-values-that-hit-zero idea shows up well beyond the textbook.
Physics and sport: a projectile's height is a quadratic in time, so its zeros are the launch and landing moments, and they set the flight time and the range.
Business: when profit is modelled as a quadratic, the zeros are the break-even points where profit is exactly zero, one on the way up and one on the way down.
Engineering: locating where a modelled quantity crosses a reference level, such as where a cable's parabolic sag meets a support line, is a search for zeros.
Astronomy: some comets and spacecraft follow near-parabolic paths, and the zeros mark where the path crosses a chosen axis.
Physics, money, structures, and orbits all lean on the same short question: where does this curve hit zero?
What Are The Most Common Zeros Of A Quadratic Polynomial Mistakes?
These four errors account for most lost marks on this topic, verified against published error guides and the ranked results for common student mistakes with zeros.
Reading the coefficients as the zeros.
Where it slips in:
Looking at $x^2 - 5x + 6$, a student writes the zeros as $5$ and $6$, or points to the constant term as a zero.
Don't do this:
The numbers $a$, $b$, and $c$ are coefficients, not zeros. The value at $x = 0$ is the constant $c$, which is the y-intercept, not a zero.
The correct way:
Set the polynomial equal to zero and solve it. For $x^2 - 5x + 6 = 0$ the zeros are $2$ and $3$, which are nowhere in the coefficient list.
Flipping the sign, or swapping sum and product.
Where it slips in:
A student writes the sum of the zeros as $\tfrac{b}{a}$, or reports the sum as $\tfrac{c}{a}$ and the product as $-\tfrac{b}{a}$.
Don't do this:
Do not drop the minus sign, and do not trade the two formulas. The sum carries the negative; the product does not.
The correct way:
Keep them fixed: sum $= -\tfrac{b}{a}$, product $= \tfrac{c}{a}$. A quick check on $x^2 - 5x + 6$ gives sum $= 5$ and product $= 6$, which matches the zeros $2$ and $3$.
Reading off $a$, $b$, $c$ before writing standard form.
Where it slips in:
Faced with $x^2 - 4x = 3$, a student reads $c = 3$ and $b = -4$ straight away, without moving every term to one side first.
Don't do this:
Do not identify coefficients from an unbalanced equation, and do not assume $a$ is always $1$.
The correct way:
Rewrite in the standard form first. Here $x^2 - 4x - 3 = 0$, so $a = 1$, $b = -4$, and $c = -3$. See standard form of a quadratic equation.
Forgetting that a negative discriminant means no real zeros.
Where it slips in:
A student computes $b^2 - 4ac$, finds a negative value, and then invents two real answers anyway.
Don't do this:
Do not force a real answer when the discriminant is negative. The square root of a negative number is not real.
The correct way:
Report that there are no real zeros. For $x^2 + x + 1$ the discriminant is $1 - 4 = -3$, so the polynomial has no real zeros and the parabola never crosses the x-axis.
Practice Problems On Zeros Of A Quadratic Polynomial
Work each one, then check against the answer that follows it.
Find the zeros of $x^2 - 7x + 12$ by factoring.
(Answer: $(x-3)(x-4)$, so the zeros are $3$ and $4$.)Use the quadratic formula to find the zeros of $x^2 - 4x + 1$.
(Answer: $x = 2 \pm \sqrt{3}$.)Without solving, find the sum and product of the zeros of $3x^2 - 5x + 2$.
(Answer: sum $= \tfrac{5}{3}$, product $= \tfrac{2}{3}$.)How many real zeros does $x^2 + 2x + 5$ have?
(Answer: discriminant $= 4 - 20 = -16 < 0$, so no real zeros.)Form a quadratic polynomial whose zeros are $-2$ and $5$.
(Answer: $x^2 - 3x - 10$.)One zero of $x^2 - 8x + k$ is $3$. Find $k$ and the other zero.
(Answer: $9 - 24 + k = 0$ gives $k = 15$; the sum of the zeros is $8$, so the other zero is $5$.)
Where Should You Go Next After Zeros Of A Quadratic Polynomial?
Zeros are the doorway into the wider world of quadratics and polynomials, and a few natural next steps build straight on what the zeros gave you.
Roots of a quadratic equation. The same values seen from the equation side, with more solving practice.
The discriminant. A closer look at the one expression that predicts how many real zeros you will get.
Zeros of a cubic polynomial. Move up to degree three, where a polynomial can have as many as three zeros.
To build this with a live trainer who starts from why the zeros matter, families can look at the Bhanzu algebra program.
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